Problem 27 - Precalculus : Building Concepts and Connections

Question

Find the exact value of each trigonometric function. $$\cos 30^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \(\cos 30^{\circ}\) is \(\sqrt{3}/2\).

Step 1: Recognition of Special Triangle

The 30-60-90 triangle is a special right triangle. The ratios of the sides of this triangle are always the same and are as follows: the length of the hypotenuse is twice the length of the shorter side, and the length of the longer side is square root of 3 times the length of the shorter side. In a 30-60-90 triangle, the cosine of 30 degrees (the shorter angle) can be found by the ratio of the adjacent side to the hypotenuse.

Step 2: Determine the Cosine Value

In the 30-60-90 triangle, the cosine of 30 degrees is the ratio of the side adjacent to the angle (which is the longer side) to the hypotenuse. The ratio is \(\sqrt{3}/2\) to \(2\). When simplified, this ratio gives the cosine value of 30 degrees which is \(\sqrt{3}/2\).

Key Concepts

Special Triangles

Special triangles in trigonometry are triangles with angles and side ratios that make calculations easier. These triangles have predictable side lengths relative to each angle, helping us find trigonometric function values like sine, cosine, and tangent quickly.

Two commonly used special triangles are:

• 45-45-90 triangle: Both of the non-hypotenuse sides are equal in length, and the hypotenuse is \( \sqrt{2} \) times one side.
• 30-60-90 triangle: This is where the hypotenuse is twice as long as the shorter side, and the longer side is \( \sqrt{3} \) times the shorter side.

Memorizing these triangles' side ratios simplifies many trigonometric problems. By recognizing situations where these triangles fit, you can calculate exact trigonometric values without a calculator.

30-60-90 Triangle

The 30-60-90 triangle is a type of right triangle where the angles measure 30, 60, and 90 degrees. Each angle has a specific and consistent relationship with the triangle's side lengths.

The side length ratios in a 30-60-90 triangle are:

• The shortest side (opposite the 30-degree angle) is known as \(x\).
• The hypotenuse (opposite the 90-degree angle) is \(2x\).
• The longer side (opposite the 60-degree angle) is \(x \sqrt{3}\).

These consistent ratios provide a straightforward way to calculate trigonometric functions. If you know one side, you can find the others using these ratios. This triangle makes it easy to solve for the cosine of 30 degrees, as it is the ratio of the sides adjacent to the 30-degree angle.

Cosine Function

The cosine function is a fundamental part of trigonometry, used to relate the angle of a right triangle to the lengths of its sides. Specifically, it is the ratio of the length of the adjacent side to the length of the hypotenuse.

In a 30-60-90 triangle, to find \( \cos 30^{\circ} \), you look at the relation:

• Adjacent side (longer side) ratio: \( x\sqrt{3} \)
• Hypotenuse ratio: \( 2x \)

So, the cosine of 30 degrees, \( \cos 30^{\circ} \), is determined by dividing the adjacent side by the hypotenuse:\[\cos 30^{\circ} = \frac{x\sqrt{3}}{2x} = \frac{\sqrt{3}}{2}.\]This calculation gives you the exact value of the cosine of 30 degrees: \( \frac{\sqrt{3}}{2} \). Understanding the cosine function in special triangles helps solve many trigonometric problems effortlessly.